Abstract: A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials

Abstract: The orthogonal polynomials that are the subject of these lectures are Laurent polynomials in several variables. They depend rationally on two parameters q and t, and there is a family of them attached to each root system R. For particular values of the parameters q and t, these polynomials reduce to objects familiar in representation theory: (i) when q = t,they are independent of q and are the Weyl characters for the root system R. (ii) when q = 0 they are (up to a scalar factor) the polynomials that give the values of zonal spherical functions on a semisimple p-adic Lie group G relative to a maximal compact subgroup K, such that the restricted root system of (G,K) is the dual root system R. (iii) when q and t both tend to 1, in such a way that (1 – t)/(1 – q) tends to a definite limit k , then (for certaion values of k) our polynomials guive the values of zonal spherical functions on a real (compact or noncompact) symmetric space G/K arising from finite-dimensional spherical representations of G, that is to say representations having a non zero K-fixed vector. Here the root system R is the restricted root system of G/K, and the parameter k is half the root multiplicity (assumed to be the same for all restricted roots).

Abstract: We survey in this paper characterization theorems dealing with polynomial sets which are orthogonal on the real line.

Abstract: Zernike polynomials have been used for some time to fit wavefront deformation measurements to a two-dimensional polynomial. Their orthogonality properties make them ideal for this kind of application. The typical procedure consists of first obtaining the fitting using x-y polynomials and then transforming them to Zernike polynomials by means of a matrix multiplication. Here, we present a new method for making this fitting faster by using a set of orthogonal polynomials on a discrete base of data points on a unitary circle.

Abstract: Orthogonal bases for the algebras of functions of Podles' quantum 2-spheres are explicitly determined in terms of bigq-Jacobi polynomials. This gives a group-theoretic interpretation of the symmetric bigq-Jacobi polynomials and the symmetricq-Hahn polynomials.

Abstract: A confluent Vandermonde-like matrix $P(\alpha _0 ,\alpha _1 , \cdots ,\alpha _n )$ is a generalisation of the confluent Vandermonde matrix in which the monomials are replaced by arbitrary polynomials. For the case where the polynomials satisfy a three-term recurrence relation algorithms for solving the systems $Px = b$ and $P^T a = f$ in $O(n^2 )$ operations are derived. Forward and backward error analyses that provide bounds for the relative error and the residual of the computed solution are given. The bounds reveal a rich variety of problem-dependent phenomena, including both good and bad stability properties and the possibility of Xextremely accurate solutions. To combat potential instability, a method is derived for computing a “stable” ordering of the points $\alpha _i $; it mimics the interchanges performed by Gaussian elimination with partial pivoting, using only $O(n^2)$ operations. The results of extensive numerical tests are summarised, and recommendations are given for how to use the fast algo...

Abstract: It is often desirable to obtain (asymptotic) properties of orthogonal polynomials and the measure with respect to which these polynomials are orthogonal. All orthogonal polynomials on the real line (with a positive Borel measure) satisfy a three term recurrence relation. We give a survey showing how properties of the recurrence coefficients reveal properties of the corresponding orthogonal polynomials.

Abstract: Matching two-sided estimates are given for the minimal degree of polynomialsP satisfyingP(0)=1 and ¦P(x)|≤exp(−ϕ (¦x¦)),x ∈ [−1,1], whereϕ is an arbitrary, in [0, 1], increasing function. Besides these fast decreasing polynomials we also consider bell-shaped polynomials and polynomials approximating well the signum function.

Abstract: The aim of this contribution is first to show how the Szego theory of orthogonal polynomials on the unit circle is intimately related to the celebrated Levinson algorithm, which is commonly used in digital signal processing (DSP) applications to solve various least-squares problems. A computationally more efficient substitute for the Levinson algorithm, termed the split Levinson algorithm, has recently been proposed in the DSP literature. In the case of real data, this new algorithm can be interpreted naturally in the framework of a well-defined one-to-one correspondence between the families of real Szego polynomials and the families of polynomials orthogonal on the interval [-1, 1] with respect to a symmetric measure. More generally, the philosophy underlying the split Levinson algorithm opens the door to an interesting “tridiagonal approach” to the theory of complex Szego polynomials, nonnegative definite Hermitian Toeplitz matrices, and related algebraic and function theoretic questions. Some of the main topics of this new mathematical framework are briefly reviewed and are shown on specific examples to be of particular interest in DSP applications.

Abstract: The problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered. A formula is proved for the ultraspherical polynomials, of which the Chebyshev and Legendre polynomials are important special cases.

Abstract: Central limit theorems are proved for random walks on the non-negative integers where the transition probabilities are homogeneous with respect to a sequence of orthogonal polynomials. Assuming some restrictions concerning the three-term recursion formula of these polynomials, one gets a Rayleigh distribution as limit distribution where bounds of the order of convergence can be computed explicitly. These central limit theorems are applied to generalized birth and death random walks and random walks on polynomial hypergroups. Finally some examples of polynomial hypergroups are discussed in view of the limit theorems above.

Abstract: The problem of generating the recurrence coefficients of orthogonal polynomials from the moments or from modified moments of the weight function is well understood for positive weight distributions. Here we extend this theory and the basic algorithms to the case of an indefinite weight function. While the generic indefinite case is formally not much different from the positive definite case, there exist nongeneric degenerate situations, and these require a different more complicated treatment. The understanding of these degenerate situations makes it possible to construct a stable approximate solution of an ill-conditioned problem. The application to adaptive iterative methods for linear systems of equations is anticipated.

Abstract: We establish a formula for the number of irreducible polynomialsf(x) over the binary fieldF2 of given degreen ≧ 2 for which the coefficient ofx n-1 and ofx is equal to 1. This formula shows that the number of such polynomials is positive for alln ≧ 2 withn ≠ 3. These polynomials can be applied in a construction of irreducible self-reciprocal polynomials overF2 of arbitrarily large degrees.

Abstract: This paper studies the measure of orthogonality for a system of polynomials defined by a three term recursion formula, using the techniques of operator theory and functional analysis. Spectral properties of self-adjoint operators and compact operators, perturbation theorems, and commutator equations are used in the development of the ideas.